Vertex Operators, Solvable Lattice Models and Metaplectic Whittaker Functions
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Communications in
Mathematical Physics
Vertex Operators, Solvable Lattice Models and Metaplectic Whittaker Functions Ben Brubaker1 , Valentin Buciumas2 , Daniel Bump3 , Henrik P. A. Gustafsson3,4,5,6 1 School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA.
E-mail: [email protected]
2 School of Mathematics and Physics, The University of Queensland, St. Lucia, QLD 4072, Australia.
E-mail: [email protected]
3 Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA.
E-mail: [email protected]
4 Present address: School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA. 5 Present address: Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA. 6 Present address: Department of Mathematical Sciences, University of Gothenburg and Chalmers University
of Technology, 412 96 Gothenburg, Sweden. E-mail: [email protected]
Received: 3 December 2018 / Accepted: 27 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: We show that spherical Whittaker functions on an n-fold cover of the general linear group arise naturally from the quantum Fock space representation of Uq ( sl(n)) introduced by Kashiwara, Miwa and Stern (KMS). We arrive at this connection by reconsidering the solvable lattice models known as “metaplectic ice” whose partition functions are metaplectic Whittaker functions. First, we show that a certain Hecke action on metaplectic Whittaker coinvariants agrees (up to twisting) with a Hecke action of Ginzburg, Reshetikhin, and Vasserot arising in quantum affine Schur-Weyl duality. This allows us to expand the framework of KMS by Drinfeld twisting to introduce Gauss sums into the quantum wedge, which are necessary for connections to metaplectic forms. Our main theorem interprets the row transfer matrices of this ice model as “half” vertex operators on quantum Fock space that intertwine with the action of Uq ( sl(n)). In the process, we introduce new symmetric functions termed metaplectic symmetric functions and explain how they are related to Whittaker functions on an n-fold metaplectic cover of GLr . These resemble LLT polynomials or ribbon symmetric functions introduced by Lascoux, Leclerc and Thibon, and in fact the metaplectic symmetric functions are (up to twisting) specializations of supersymmetric LLT polynomials defined by Lam. Indeed Lam constructed families of symmetric functions from Heisenberg algebra actions on the Fock space commuting with the Uq ( sl(n))-action. The Heisenberg algebra is independent of Drinfeld twisting of the quantum group. We explain that half vertex operators agree with Lam’s construction and this interpretation allows for many new identities for metaplectic symmetric and Whittaker functions, including Cauchy identities. While both metaplectic symmetric functions and LLT polynomials can be related to vertex operators on the quantum Fock space, only metaplectic symmetric functions are connected to solvable lattice models.
B. Brubaker, V. Buciumas, D.
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